Can More Gender Equality Lead to Higher

Fertility?

Nils-Petter Lagerlof∗

Department of Economics

York University

4700 Keele Street

Toronto M3J 1P3

Canada

E-mail: lagerlof@econ.yorku.ca

September 9, 2003

∗This and earlier papers in this project have beneÞted from discussions with Gautam Bose,

Buly Cardak, Carmen Carrera, Stephan Klasen, Paul Klein, Jan Ondrich, Claus Poertner, and

Efraim Sadka, as well as participants at the Economic Theory Workshop in Melbourne 1998,

the ESPE Meeting in Turin 1999, and seminar participants at Concordia University, University

of New South Wales, and RSSS, Australian National University.

Abstract: We set up a two-sex overlapping-generations model with endoge-

nous fertility. Due to a coordination process across houdeholds, parental human-

capital investment may differ between sons and daughters. In societies with plenty

of gender discrimination children are cheap to rear, since women�s time is cheap,

so they tend to have large families, invest little human capital in each child, and

have stagnant incomes. Equal societies tend to have small families and growing

levels of income. However, the relationship between gender equality and fertility

among non-growing economies is the exact opposite: among countries too unequal

to exhibit sustained growth, those with relatively with more gender equality have

higher steady-state fertility.

Keywords: Gender equality, gender gap, fertility, growth.

JEL classification codes: D91, J13, J16, O11.

2

1. Introduction

Economic development seems connected to the status of women. Whether mea-

sured by gender differences in education and life expectancy, indexes of legal rights

for women, or percentage of women in parliament, gender equality is positively

correlated with per-capita income across countries [see Dollar and Gatti (1999)].

The picture is similar when looking at trends in schooling for different regions of

the world. The ratio of female years of schooling over that of males has been con-

stant at around .95 in the OECD world 1960-90. In developing countries this ratio

has been considerably lower, but it has increased from .56 in 1960, to .69 in 1990.

East Asian and PaciÞc countries, however, have displayed a more emphasized

increase, from .59 in 1960 to .84 in 1990 [Barro and Lee (1993, 1996)].

Interestingly, those regions which have experienced the sharpest increase in

gender equality (such as OECD the past century1 and East Asia more recently)

are the very same ones which have escaped poverty. Moreover, this has been

paralleled by a fall in fertility rates. Thus, it is often argued that educating

women could be a key to economic development, and a break on the population

explosion [see e.g. Sen (1997) and Summers (1998)]. The issue has also inspired

an empirical growth literature [see e.g. Barro and Lee (1993, 1994, 1996), Stokey

(1994), Hill and King (1995), Dollar and Gatti (1999), Klasen (2002), and Knowles

et al. (2001)].

However, most of these empirical studies fail to provide any explicit theoretical

1See Olsen (1994) for the U.S.

3

model of growth and gender heterogeneity, so it is not clear how to interpret the

regressions. Also, any convincing interpretation should be consistent with micro

evidence on how gender equality affects fertility: namely that fertility is increasing

with the income of the male spouse, but falling with that of the female spouse.

[See Butz and Ward (1979), Heckman and Walker (1990), and Schultz (1995).]

This paper builds on work by Lagerlof (2003). We set up a two-sex overlapping-

generations model, where fertility is endogenous, and subject to both a goods

cost, and a time cost. There is also gender discrimination, expressed by parents�

division of human-capital investment between sons and daughters. Our results

are consistent with the mentioned micro evidence: fertility increases with male

income, but falls with female income. They also broadly match the macro evidence

described above. In a society with much gender discrimination children are cheap,

since women�s time is cheap. So unequal societies tend to have large families, and

invest little in quality (i.e., human capital). As a result, they are less likely to

exhibit sustained growth in human capital, and more likely to end up in a poverty

trap, with non-growing levels of income. Equal societies tend to have small families

and growing levels of income. Also, among growing economies, the more equal

ones have lower fertility and grow faster.

The novelty here compared to Lagerlof (2003) is twofold. First, we focus

only on cross-country patterns in the world today, and not on changes in gender

equality over long periods of time, such as those associated with the birth and

spread of Christianity.

4

Second, we use a general equilibrium setting, where factor prices are deter-

mined as the marginal products of labor and capital. This complicates the analy-

sis a great deal, due to an interdependence of the wage rate and fertility. Among

growing economies, the wage rate converges to an expression which we can solve

for analytically, and we get the standard result of more gender equality leading

to lower fertility (and more growth). More surprisingly, the relationship between

gender equality and fertility among non-growing countries is exactly the oppo-

site. Countries with more gender equality (although still too unequal to exhibit

sustained growth) have higher fertility than the less equal.

The Þrst part of the intuition is similar to that of Lagerlof (2003) and comes

from fertility being costly in both time and goods. The goods cost can be thought

of as spending on food and clothes, which becomes less important as income levels

grow over time, but matters to economies stuck in a poverty trap. The time cost,

which measures the cost of foregone income, matters most to rich and growing

economies, but less to poor. A rise in female income relative to that of males

induces lower fertility, in poor countries as well as rich, but the associated increase

in human capital investment in children raises the income of the next generation.

Therefore, the next generation (men and women alike) becomes richer compared

to their parents. Since this is still a poor economy, where the goods cost of children

matters, a proportional increase in income of both parents induces higher fertility.

As the economy stabilizes in a new non-growing steady state, fertility turns out

to be proportional to the wage rate.

5

The second part of the intuition has to do with the general equilibrium frame-

work. In Lagerlof (2003) the wage rate is exogenous and constant, so fertility is

unchanged in the new steady state. In the setting used here the wage rate in the

new steady state is higher, and so is fertility.

Earlier attempts to endogenize fertility in growth models date back over a

decade [see Becker and Barro (1988), Barro and Becker (1989), and Becker et

al. (1990)]. Barro and Becker (1989) and Barro and Sala-i-Martin (1995, Ch. 9)

show that the combination of both a time and goods cost can lead to differences

in fertility behavior between poor and rich countries, using general equilibrium

frameworks, but do not study the implications of an explicit gender gap in income.

Attempts to explicitly allow for gender heterogeneity in growth models are

relatively recent.2 Most two-sex models assume exogenous gender differences:

Davies and Zhang (1995) put son-preferences in the utility function of the parents.

Galor and Weil (1996) assume two types of labor: �brains,� of which men and

women have equal amounts, and �brawns,� with which only men are endowed.

Elul et al. (1997) assume that men marry younger women. Echevarria and Merlo

(1999) assume that women bear children at a certain biological time cost, and

thus work less than their spouses. Dasgupta (2000) lets production take place

in an informal and a formal sector, and assumes that men can enter the formal

sector directly, whereas women have to spend some time in the informal sector

2For an overview, as well as a good motivation for the two-sex approach, see Echevarria and

Moe (2000).

6

before being able to enter the formal sector.3

The way we model gender discrimination relies on no exogenous differences

between men and women, except that of heterosexuality : men are matched with

women, and women with men. We assume that a parent couple cares about the

total income of the households into which their children enter as they grow up.

Therefore, not only the offspring�s income matter, but also that of his/her future

spouse, who is educated by some other family. Families thus play a (normal form)

coordination game against each other. From the viewpoint of one atomistic parent

couple, a daughter (son) may not need much education, simply because she (he)

is expected to marry a man (woman), who � taking the equilibrium behavior

of other families as given � may be better educated, and earn a higher income.

Gender discrimination can thus arise in a Nash equilibrium, despite the sexes

being symmetric. In that sense, our model captures the concept of gender roles,

or gender stereotypes : if everyone else behaves in a discriminatory manner, it is

optimal for the atomistic player to do the same.4

3In a related but different context, Cole et al. (1992) examine the interaction between growth

and status, the latter being a ranking device determining marriage outcomes. Most of the other

papers link growth and gender heterogeneity via the fertility decision, whereas Cole et al. (1992)

study the capital accumulation coming from bequests, and the way they interact with status.4This explains why we observe educational discrimination on the basis of gender, but not why

women in particular are discriminated against. We do not argue that �biology� does not matter.

In fact, coordination can amplify (small) biological differences between the sexes. See the

discussion in HadÞeld (1999) and Francois (1998) who model similar coordination mechanisms

in different contexts. Nerlove et al. (1984) analyze similar coordination in a one-sex model.

7

Modelling gender inequality in this way gives some indirect policy implications.

Since gender inequality, and thus poverty, originates from coordination, a shift

to sustained growth could potentially be achieved by re-coordinating on a less

discriminatory equilibrium. A tentative conclusion is therefore that campaigns

to mitigate the gender bias in education may enhance growth. This relates to

the idea of a �cooperative� way of reducing fertility rates and boosting economic

development, as suggested by e.g. Sen (1997).5

The rest of this paper is organized as follows. Section 2 sets up the model. It

describes the composition of the cost of rearing children, sets up the maximiza-

tion problem of each couple, derives the Nash equilibrium in each period. Then

the production side is described and the dynamics analyzed, Þrst for growing

economies, then for non-growing. Finally, Section 2 analyzes the dynamics and

stability properties. Section 3 gives a concluding summary.

2. The Model

The structure of this model is similar to that in Lagerlof (2003). Individuals live in

overlapping generations for three periods: childhood, adulthood (or working age),

and retirement (or old age). Individuals are either males or females and when

entering adulthood they marry randomly into pairs. The sexes have identical

abilities at birth, but may be endowed with different amounts of human capital

5This type of policy implication does not arise in most other frameworks, such as that of Galor

and Weil (1996), who let gender inequality be generated by (exogenously assumed) differences

in the types of labor men and women are endowed with.

8

by their parents before marriage.

A period-t adult male is endowed with hmt units of human capital, and a female

in the same generation is endowed with hft units of human capital. There are no

differences within the sexes: men belonging to the same generation have equal

amounts of human capital, and the same for women. Each unit of human capital

earns a wage, wt, per unit of time supplied on the labor market. Different from

the case in Lagerlof (2003) this wage rate is endogenous. The wage is the same for

both sexes, so there is in that sense no gender discrimination on the labor market.

A couple formed by adults in period t is referred to as couple t. Given their

joint income, wt(hmt + h

ft ), couple t decides how many children to have, nt, half

of whom are daughters, and half are sons. They also endow each child with hit+1

(i = m, f) units of human capital, where the super-indices indicate the human

capital of a daughter (f), and a son (m), respectively. In a standard fashion, we

talk about fertility, nt, as quantity of children, and human capital, hmt+1+h

ft+1, as

quality.

2.1. The child rearing cost

To rear children carries three types of cost: a time cost, a consumption goods

cost, and the cost of buying human capital to invest in the children. Let the sum

of the Þrst two components be denoted qt.

Consider a couple active in period t, where (for whatever reason) the man

has more human capital than the female, hmt > hft . If they chose to have few

enough children only one spouse�s time is needed for their upbringing. (That this

9

is indeed the case will be guaranteed by restricting attention to a certain set of

Nash equilibria; see below.)

Let the time cost be b units of time per child, and the goods cost a units of

the consumption good per child. The female spouse has lower human capital and

earns less than the male so she undertakes the whole time cost. She earns wthft

per unit of time and spends b units of time per child; the couple thus foregoes

bwthft in lost income per child. The sum of the goods and time cost per child is

thus given by

qt = a+ bwthft . (1)

2.2. Consumption and saving

The parent couple acts as one individual, and care about the (potential) income

earned in the households into which their children enter. This is given by the

wage rate in the next period, wt+1, times the sum of the human capital levels

of their own children, and that of their sons- and daughters-in-law. The parents

care about their own consumption (in old and working age), and the number of

children they have. We assume that all these goods are collective to the couple,

abstracting from intra-couple allocation conßicts. The utility function is given by

Ut = ln(c1t) + α ln(nt) + δ ln(c2t+1)

+βhlnhwt+1(h

mt+1 + h

f

t+1)i+ ln

hwt+1(h

m

t+1 + hft+1)

ii,

(2)

10

where c1t and c2t+1 is consumption in working and old age, respectively. Recall

that hft+1 and hmt+1 denote the human capital of couple t�s daughters and sons,

respectively. The same variables with a bar denote the human capital of his/her

spouse, which is the same as the human capital of any man or woman belonging

to the same generation as their children. The wage times the sum of these two

constitutes the earnings of the households into which the children enter in period

t + 1. (More precisely, this is the potential earnings, since the actual earnings

depend on the number of children, due to the time cost.)

α, β, and δ are the weights put on fertility, children�s quality, and old-age

consumption, respectively. We have normalized the weight on working-age con-

sumption to unity. We assume the following, which is explained later:

Assumption 1.

1 + 2β + δ > α > β. (3)

2.3. Budget constraints

Working age consumption of couple t is given by

c1t = wt(hmt + h

ft )− st − nt

hqt + p(h

mt+1 + h

ft+1)

i, (4)

where st denotes couple t�s saving, and 2p is the (exogenous) cost of acquiring one

unit of human capital in terms of the consumption good.6 As described above, qt

is the time and goods cost of rearing one child [see (1)].

6The couple has nt/2 sons and invests hmt+1 in each, at a price of 2p. Thus, the total cost of

human capital expenditure on sons is ntphmt+1, and similarly for daughters. This gives (4).

11

Old-age consumption is given by

c2t+1 = st(1 + rt+1), (5)

where rt+1 denotes the real interest rate on savings held from period t to period

t+ 1.

2.4. Utility maximization

Maximizing (2), subject to (4) and (5), the Þrst-order conditions for st and nt say

that

st =δwt(h

mt + h

ft )

1 + α + δ, (6)

and

nt =αwt(h

mt + h

ft )

(1 + α + δ)hqt + p(hmt+1 + h

ft+1)

i . (7)

The Þrst-order conditions for hmt+1 and hft+1, taking h

f

t+1 and hm

t+1 as given, can

be written:7

βhhmt+1 + h

f

t+1

i−1

≤ αhqt + p

³hmt+1 + h

ft+1

´i−1

,

βhhm

t+1 + hft+1

i−1

≤ αhqt + p

³hmt+1 + h

ft+1

´i−1

.

(8)

The weak inequalities are due to the non-negativity constraint on human capital.

If hmt+1 > 0 in optimum, the Þrst weak inequality in (8) holds with equality, and

7This can be seen by substituting (4), (6) and (7) into the objective function in (2) and

taking Þrst-order conditions with respect to hmt+1 and hft+1.

12

similarly for the second weak inequality if hft+1 > 0. If hmt+1 is constrained to zero,

the Þrst inequality is strict, and vice versa with the second inequality if hft+1 is

constrained to zero.

The left-hand sides capture the marginal beneÞt to the atomistic parent couple

of endowing their children with human capital. The higher is the income of the

future spouses of their children (hf

t+1 and hm

t+1), the lower is the marginal utility

for the atomistic parent couple of increasing their children�s family income. Thus,

a high average level of human capital among men induces the atomistic parent

couple to invest little in their daughter�s human capital, and vice versa if the

average human capital level of women is high.

The right-hand sides of (8) give the marginal cost of increasing human capital

investment in children, as implied by the associated reduction in fertility, as follows

from the budget constraint (4).

2.5. Nash equilibrium

The players of the (normal form) game are the parent couples, and they choose

simultaneously how much to invest in sons and daughters, taking as given the

same choices by the other parents. In a (symmetric pure strategy) Nash equilib-

rium there are no differences within the sexes, so hmt+1 = hm

t+1 and hft+1 = h

f

t+1.

This makes the two conditions in (8) coincide, and therefore the non-negativity

constraints cannot be binding: the Þrst-order conditions in (8) hold with equality.

Moreover, we see that the sum of male and female human capital can be written

as

13

hmt+1 + hft+1 =

µβ

α− β¶qtp. (9)

In fact, this demonstrates that in every period t there is a continuum of Nash

equilibria.

Remark 1. Any combination of hmt+1 and hft+1 which satisfies (9) is a Nash equi-

librium (as long as they are both non-negative).

Note also from the Assumption 1 that α > β ensures that (9) is positive, which

amounts to assuming an interior solution. If α approached β the couple would

tend to choose an arbitrarily small number of children, investing an arbitrarily

large amount of human capital in each (fraction of a) child.

2.6. Gender equality

DeÞne the following measure of gender equality:

µt =hfthmt. (10)

As follows from Remark 1, any non-negative level of µt is a Nash equilibrium,

as long as (9) holds (lagged one period). There is nothing linking gender equality

in one period to that in the next. Any sequence of (non-negative) µt�s is consistent

with a Nash equilibrium in the game played between different parent couples in

each respective period. However, later in the paper we shall assume that gender

equality does not change over time. In other words, we are going to assume that

14

parent couples in period t coordinate on the same Nash equilibrium as their parent

generation t− 1 did.

We shall only consider Nash equilibria where

µt ∈ (α− β

1 + β + δ, 1). (11)

The upper bound in (11) simply serves to focus our attention on the empirical

regularity that women tend to have lower human capital than men, hmt > hft .

(However, as long as there are no exogenous differences between the sexes, there

is always a mirror imaged equilibrium in which men have lower human capital.)

The lower bound serves to ensure that the couple is equal enough so that (on

those growth paths we are to consider) the couple has few enough children that

only one spouse�s time is needed to take care of them. [See Appendix A.2.] If

this was not the case, the marginal cost of having an extra child would make a

discrete jump at some fertility level, where the father started making a positive

time input. Note that Assumption 1 implies that the lower bound is less than

unity.

2.7. Fertility

Using (9) and (7) we can write fertility as

nt =

µα− β

1 + α + δ

¶wt(h

mt + h

ft )

a+ bhftwt=

µα− β

1 + α + δ

¶wt(1 + µt)ahmt+ bwtµt

, (12)

where the second equality comes from (10).

15

2.8. Production

In period t, a representative Þrm produces output (Yt) of the single consumption

good, using a constant returns to scale production function of Cobb-Douglas type,

with inputs physical capital (Kt), and efficiency units of labor (Lt). Letting ρ

denote the capital share of output we can write the production function as

Yt = Kρt L

1−ρt . (13)

The total number of couples in working age in period t is denoted Nt, so that the

working force in terms of efficiency units is given by

Lt = Nthhmt + h

ft (1− bnt)

i. (14)

(Recall that we are considering equilibria where hmt > hft , so that all time spent

rearing children is levied on the female.)

Letting lower case letters denote per-couple terms of all variables, given by

dividing them by Nt, we can write the production function as

yt = kρt

hhmt + h

ft (1− bnt)

i1−ρ. (15)

ProÞt maximization equates the factor prices to their respective marginal prod-

ucts, in a standard fashion, so that

wt = (1− ρ)·

kt

hmt + hft (1− bnt)

¸ρ, (16)

16

and similarly for rt+1, which turns out to play no role in the rest of the analysis,

due to the logarithmic utility.

2.9. Dynamics

Each of the Nt couples has nt/2 children of each sex, and it takes one offspring of

each sex to form a new couple. Therefore the number of couples evolves according

to Nt+1 = (nt/2)Nt. Since physical capital is made up of the previous period�s

saving the per-couple physical capital stock evolves according to

kt+1 =stnt/2

=

µ2δ

α− β¶ha+ bhftwt

i, (17)

where the second equality comes from (6), and (12). Using the notation in (10),

i.e., µt = hft /hmt , and (9), we can write the ratio of physical capital, over male

human capital, as

kt+1

hmt+1

=2δ(1 + µt+1)

pβ. (18)

Lagging (18) one period, using (10) again, the wage rate in (16) can be written

as

wt = (1− ρ) 2δ/(βp)

1− b³

µt1+µt

´nt

ρ ≡ ω(nt;µt). (19)

The economy may exhibit sustained growth, or converge to a non-growing

steady state: a poverty trap. We now examine under which conditions each sce-

nario occurs. Sustained growth in output requires that both human and physical

17

capital exhibit sustained growth. Since their ratio is constant [see (18)] we can

look at the growth of human capital.

Let the ratio µ = hft /hmt be constant over time. This amounts to assuming

that each economy coordinates on the same Nash equilibrium over time. [See the

discussion below (10).] From (1) and (9), we see that the growth rate of male

human capital hmt (as well as the total human capital of each household hmt + h

ft )

is given by

γt+1 =hmt+1

hmt=

hmt+1+hft+1z }| {hmt+1(1 + µ)

hmt (1 + µ)=

µβ/p

α− β¶" a

hmt+ bµwt

1 + µ

#. (20)

2.9.1. Growing economies

The economy exhibits sustained growth if the right-hand side of (20) approaches

something greater than unity as hmt approaches inÞnity. To examine when this is

the case we must determine the limit of wt as hmt approaches inÞnity. Start by

looking at the asymptotic behavior of fertility. From (12) we see that

limhmt →∞

nt ≡ n∗ =α−β

1+α+δ

b³

µ1+µ

´ . (21)

(Recall that we are holding the ratio µ = hft /hmt constant so h

ft approaches inÞnity

together with hmt .) Using (21) and (19), the asymptotic wage rate can be written

limhmt →∞

wt ≡ w∗ = (1− ρ)·µ1 + α + δ

1 + β + δ

¶µ2δ

βp

¶¸ρ, (22)

which gives the asymptotic growth rate

18

limhmt →∞

γt+1 ≡ γ∗ =µ(β/p)(1− ρ)α− β

¶·µ1 + α + δ

1 + β + δ

¶µ2δ

βp

¶¸ρµbµ

1 + µ

¶. (23)

If the right-hand side of (23) exceeds unity the economy will exhibit sustained

growth, at the (gross) rate γ∗; if it falls below unity the economy converges to a

non-growing poverty trap. [An illustration is given later in Figure 3.] Since the

factor µ/(1 + µ) in (23) in increasing in µ, we conclude the following:

Result 1. An economy with more gender equality (greater µ) is more likely to

exhibit sustained growth.

Result 2. Among economies exhibiting sustained growth, the growth rate is

higher for those with more gender equality (greater µ).

2.9.2. Non-growing economies

Now consider economies for which the right-hand side of (23) falls below unity. We

want to Þnd out how the fertility rate among such economies varies with gender

equality (given that gender equality is low enough to rule out sustained growth).

One problem now arises: for Þnite hmt the fertility rate cannot be determined

explicitly. We must instead determine the fertility and wage rates implicitly and

jointly. To do this, start by rewriting (12) as

wt =a

hmt

nt/(1 + µt)¡α−β

1+α+δ

¢− b³ µt1+µt

´nt

≡ σ(nt, hmt ;µt). (24)

19

Setting (24) and (19) equal to each other, we get wage and fertility rates which

are consistent with a labor market equilibrium. This is illustrated in Figure 1,

where σ(nt, hmt ;µt) is given in (24), and ω(nt;µt) in (19).

The graph of σ(nt, hmt ;µt) shows the demand for children. A higher wage

implies both a higher income, but also a higher time cost of children. As long

as the child rearing cost contains a positive consumption goods element (a > 0)

an increase in the wage rate does not increase the time cost as much as income.

Therefore the demand for children increases with the wage rate. If a = 0 fertility

equals n∗ independently of the wage rate, due to the logarithmic utility.

The graph of ω(nt;µt) shows how the wage, given by the marginal product of

labor, increases as fertility increases and female labor supply falls.

As hmt grows (at a given µ) the graph of σ(·) shifts down and the fertility rateincreases. If hmt grows without bound nt approaches n

∗. [Since a/hmt simultane-

ously approaches zero (24) does not tell us that the wage rate approaches inÞnity

as nt approaches to n∗.]

The graphs of σ(·) and ω(·) both have positive slope, so they may intersect

more than once: there may exist more than one labor market equilibrium. In-

tuitively, a high wage rate implies a high demand for children and therefore low

female labor supply. This in turn makes the marginal product of labor high, sus-

taining the high wage rate. In the rest of the paper we are not going to explore this

possibility of multiple equilibria. To ensure uniqueness we impose the following

parametric restriction:

20

Assumption 2. µα− β

1 + α + δ

¶>

4ρ

1− ρ . (25)

We can now state the following, which is proven in Appendix A.1.

Proposition 1. Under Assumption 2, in every period t there exists a unique

equilibrium fertility rate nt, for any (non-negative) levels of hmt and µ.

In other words, if the relationship between the exogenous parameters (ρ, α, β,

and δ) is such that (25) holds, the graphs of σ(·) and ω(·) intersect only once, so

there is only one labor market equilibrium at a given level of hmt and µ.

Fertility in non-growing economies. Consider a steady-state level of hmt , at

which the right-hand side of (20) equals unity. Denote this by (hm)0, and the

associated wage rate by w0. These are given by

µβ/p

α− β¶·

a/(hm)0 + bµw0

1 + µ

¸= 1. (26)

We next use (24) to write w0 as a function of the associated fertility rate n0:

w0 = σ(n0, (hm)0;µ) =a

(hm)0

n0/(1 + µ)¡α−β

1+α+δ

¢− b³ µ1+µ

´n0

. (27)

Then, using (26) to substitute (hm)0 away some algebra tells us that

w0 =

·1 + α + δ

β

¸pn0. (28)

21

That is, the fertility rate in a non-growing steady state is proportional to the

equilibrium wage rate. If the wage rate is higher in the steady state associated

with a higher µ, the same must hold for the fertility rate. We thus need to look

at the joint determination of the fertility and wage rates, by using ω(n;µ) in (19),

and (28) above, and see how they vary with µ.

The determination of the steady-state fertility and wage rates of a non-growing

economy is shown in Figure 2. It contains the graphs of (28), and ω(n;µ) in (19).

Note that they intersect twice, but the steady-state fertility rate n0 is given by the

lower intersection. It can be seen that the fertility rate at the upper intersection

exceeds n∗ deÞned in (21).8 This cannot be the steady-state fertility rate, since

for any Þnite level of hmt [like (hm)0], and any given µ, fertility must fall below

the associated n∗ (see Figure 1).

As shown, an increase in gender equality (from µ to µ0 in Figure 2) shifts up the

ω(n;µ)-graph, and the new intersection is associated with a higher steady-state

fertility rate. We can thus state the following result:

Result 3. Among non-growing economies, those with more gender equality (greater

µ) have higher steady-state fertility rates.

In Figure 2 it may look as if non-growing economies have lower fertility rates

than growing economies, since n0 falls below n∗. However, n∗ here refers to

the fertility rate of a non-growing economy, had it (counterfactually) exhibited

8To see this recall that the right-hand side of (23) must fall below unity for the economy to

not be be growing. Using (21) and (19) we see that this implies thath

1+α+δβ

ipn∗ > ω(n∗;µ).

22

sustained growth. Whether, or not, a country exhibits sustained growth depends

on gender equality, which also determines the fertility rate. In other words, non-

growing countries can have higher fertility rates because they have less gender

equality (lower µ), which is also the reason why they are not growing in the Þrst

place.

2.10. Stability

So far we have examined the behavior of economies on a sustained growth path,

or at a steady-state equilibrium, respectively. For completeness we examine how

these economies evolve dynamically around their respective equilibria. For in-

stance, we want to conÞrm that the equilibria are stable.

Using (7) and (9) we can rewrite the growth rate in (20) as

γt+1 =

µ1

p

¶µβ

1 + α + δ

¶wtnt. (29)

In Figure 1 we can interpret wt/nt as the slope of a line from the origin to the

intersection point. As hmt increases and shifts σ(·) down, the intersection moves

along the graph of ω(·). Since σ(·) must always intersect ω(·) from below, wt/nt

is falling in hmt . From (29) we see that this must also hold for the growth rate

γt+1:

∂γt+1

∂hmt=

µ1

p

¶µβ

1 + α + δ

¶ ∂ hwtnt

i∂hmt| {z }<0

< 0. (30)

23

This is illustrated in Figure 3. At a given level of gender equality (µ) the

growth rate is falling in initial male human capital. This is the mirror image

of the increasing demand for children as male income rises: as male earnings

rise parents can afford more children, and thus spend less on quality, implying

lower human capital investment in each child. This rise in fertility may seem to

contradict the empirical pattern of falling fertility rates in the developed world the

last hundred years. However, we must recall that gender equality is held constant

here. If gender equality improves the growth function shifts, as shown in Figure

3. This can cause a simultaneous (temporary) fall in fertility, and rise in growth

rates.

3. Conclusions

This paper sets up an overlapping-generations model with gender heterogeneity,

and heterosexual matching. Parents invest in their sons� and daughters� education,

taking into account that the children are going to form households with someone

of the opposite sex when they grow up. Gender discrimination may arise, because

if other households discriminate between human-capital investment in sons and

daughters, it is optimal for the atomistic household to do the same. The reason

is that a parent couple knows that their daughter will marry a man, so the �en-

dowment� of income in her future household is going to be relatively big, and vice

versa for sons. Discrimination thus occurs as a Nash equilibrium in a coordination

game, despite the absence of any differences in abilities between the sexes.

24

This model is an extension of that used by Lagerlof (2003). The implications of

gender equality for growth and development originates from the quantity-quality

trade-off in the fertility choice. Different from Lagerlof (2003) we use a general

equilibrium setting, where the wage rate is endogenous, and depends in labor

supply, which in turn depends on fertility rates.

Some results and mechanisms are similar to those in Lagerlof (2003). In par-

ticular, changes in gender equality have different implications for rich clubs com-

pared to poor. SigniÞcant increases in gender equality in our model can push poor

countries to sustained growth, but small increases in gender equality may not.

The result which contrasts most with Lagerlof (2003) is that the new and more

gender-equal non-growing steady state has higher fertility than the steady state

that prevailed before the increase in gender equality. Fertility is proportional to

the wage rate in both settings, but only when the wage rate in endogenous � as

it is here, but not in Lagerlof (2003) � do we get the result that fertility is higher

in the new steady state. In Lagerlof (2003) the wage rate was exogenous so the

fertility rate was unchanged in the new steady state.

Many possible theoretical extensions have been left out here, but could be

interesting to explore in future work. For instance, just as in Nerlove et al. (1984),

we assume that parents must choose their children�s education before they know

who their children marry. In our setting, parents have some more information:

they know that their children are going to marry someone of the opposite sex,

and this gives rise to gender discrimination. An interesting extension would be

25

to let parents also know that their children will marry someone who is about

equally well educated. Sending a daughter to college not only increases her future

income, but possibly also that of her future spouse. To explore this one would

need to relax the crucial assumption that there is no heterogeneity within sexes,

only between them.

There are other potential extensions: as in any game with multiple pure strat-

egy Nash equilibria it would be of interest to examine mixed strategy equilibria

as well. However, this turns out to be quite complex. For instance, due to the

fact that fertility is endogenous, the number of spouses with certain character-

istics will depend on the actions between which the players (the parent couples)

randomize. In a static setting, this could potentially be overcome, but at the cost

of the ability to study e.g. poverty traps.

A. Appendix

A.1. Proof of Proposition 1.

To make the proof easy to follow, we are going to derive a couple of lemmas along

the way. First let D(nt, hmt ) be the difference between ω(·) and σ(·), in (19) and

(24):

D(nt, hmt ) = (1− ρ)

2δ/(βp)

1− b³

µ1+µ

´nt

ρ − a

hmt

nt/(1 + µ)¡α−β

1+α+δ

¢− b³ µ1+µ

´nt

. (I)

The interpretation of (I) is that, given some value of hmt , the fertility rate which

26

is consistent with a labor market equilibrium must be such that D(nt, hmt ) = 0.

We Þrst note that, for every hmt , there exists some equilibrium fertility rate:

Lemma 1. For every hmt there exists some bn ∈ (0, n∗) for which D (nt, hmt ) = 0.

Proof. From (21) and (I) we have

limnt→n∗

D(nt, hmt ) = −∞, (II)

and

D(0, hmt ) = (1− ρ)(2δ/βp)ρ > 0. (III)

From the continuity of D (nt, hmt ) it must be zero at some nt ∈ (0, n∗).

Next, let bh : (0, n∗)→ R++ be such that D³nt,bh(nt)´ = 0, i.e.,

bh(nt) =·

ant/(1+µ)

( α−β1+α+δ )−b( µ

1+µ)nt

¸(1− ρ)

·2δ/(βp)

1−b( µ1+µ)nt

¸ρ . (IV)

This means that, for any bn ∈ (0, n∗) there exist some level of hmt = bh(bn) atwhich bn is consistent with a labor market equilibrium, i.e., D ³bn,bh(bn)´ = 0.Let D1 (·) denote the partial derivative of D (·) with respect to its Þrst argu-

ment. We are now ready to state the following lemma.

Lemma 2. Consider any bn ∈ (0, n∗). If, for any such point bn,

D1

³bn,bh(bn)´ < 0, (V)

then nt = bn is the unique solution to D³nt,bh(bn)´ = 0.

27

Proof. D1

³bn,bh(bn)´ < 0 and D(bn,bh(bn)) = 0 together imply that whenever

D³nt,bh(bn)´ intersects the nt axis it must do so from above; it cannot intersect

it from below. From the continuity of D(·) it follows that the intersection must

be unique.

What we need to investigate is thus whether, or not, (V) holds. Under the

parametric condition in Assumption 2 we shall see that it does.

To see this, we Þrst make the algebra somewhat simpler to handle by rewriting

D(·) in (I) as follows (for convenience, we rename the arguments):

D(x, y) = A [1− Bx]−ρ − Ey

·x

G−Bx¸, (VI)

where

A = (1− ρ)µ2δ

βp

¶ρ, (VII)

B = b

µµ

1 + µ

¶, (VIII)

E =a

1 + µ, (IX)

and

G =

µα− β

1 + α + δ

¶. (X)

28

After some algebra, we see that the (partial) derivative of D(·) with respect

to its Þrst argument becomes

D1(x, y) = ρBA [1−Bx]−(1+ρ) − Ey[G−Bx]−2

[G−Bx]− x[−B]| {z }=G

. (XI)

Evaluating D1(x, y) at y = bh(x), where D(x, y) = 0 in (VI), we see thatD1(x,bh(x)) = A [1−Bx]−ρ · ρB

1− Bx −G

x(G−Bx)¸. (XII)

From (XII), we see that D1(x,bh(x)) < 0 whenρB [x(G−Bx)] < G [1−Bx] , (XIII)

or

P (x) ≡ x2 −·(1 + ρ)G

ρB

¸x+

·G

ρB2

¸> 0. (XIV)

[It can be conÞrmed that the denominators within square brackets in (XII)

are indeed both strictly positive, since x < n∗, given by (21).] What remains to

prove is that, if (25) holds, P (x) > 0. To see this, note that P 00(x) > 0, and it

has a global minimum at x = (1−ρ)G2ρB

. Thus, if Ph

(1−ρ)G2ρB

i> 0, P (x) > 0 always

holds. Using (X) and (XIV), we see that this amounts to

α− β1 + α + δ

>4ρ

1− ρ ,

which is Assumption 2.

29

A.2. The lower bound on gender equality

In (11) we restricted attention to certain Nash equilibria. The restriction that

µ < 1 is obvious. We now explain why we cannot let µ fall below (α−β)/(1+β+δ).

Recall that, at any given level of µ, nt is increasing in hmt , and as h

mt approaches

inÞnity, nt approaches n∗. Thus, nt cannot exceed n∗. To ensure positive female

labor supply at all hmt it suffices to show that the fertility rate n∗ implies positive

labor supply by the female spouse, i.e., 1− bn∗ > 0. Use (21) to see that:

bn∗ = b

α−β1+α+δ

b³

µ1+µ

´ < 1. (XV)

If µ > (α − β)/(1 + β + δ) it is easily seen that (XV) holds. This gives the

lower bound in (11).

References

[1] Barro, R., 1997, Economic growth and convergence - a cross-country empir-

ical study, the MIT Press, Cambridge, Massachusetts.

[2] Barro, R. and G.S. Becker, 1989, Fertility choice in a model of economic

growth, Econometrica 57, 481-501.

[3] Barro R., and J.W. Lee, 1993, International comparisons of educational at-

tainment, Journal of Monetary Economics 32, 363-394.

[4] Barro R., and J.W. Lee, 1994, Sources of economic growth, Carnegie-

Rochester Conference Series on Public Policy 40, 1-46.

30

[5] Barro R., and J.W. Lee, 1996, International measures of schooling years and

schooling quality, American Economic Review 86, 218-223.

[6] Barro, R., and X. Sala-i-Martin, Economic Growth, McGraw-Hill, New York.

[7] Becker, G.S., 1991, A treatise on the family - enlarged edition, Harvard

University Press, Cambridge, Massachusetts; London, England.

[8] Becker, G.S. and R.J. Barro, 1988, A reformulation of the economic theory

of fertility, Quarterly Journal of Economics 103, 1-25.

[9] Becker, G.S., K.M. Murphy and R. Tamura, 1990, Human capital, fertility,

and economic growth, Journal of Political Economy 5, 12-37.

[10] Butz, W.P., and M.P. Ward, 1979, The emergence of countercyclical U.S.

fertility, American Economic Review 69 (3), 318-328.

[11] Cole. H.L., G.J. Mailath, and A. Postlewaite, 1992, Social norms, savings

behavior, and growth, Journal of Political Economy 100, 1092-1125.

[12] Dasgupta, I., 2000, Women�s employment, intra-household bargaining, and

distribution: a two-sector analysis, Oxford Economic Paper 52, 723-744.

[13] Davies, J.B., and J. Zhang, 1995, Gender bias, investments in children, and

bequests, International Economic Review 36 (3), 795-818.

[14] Diamond., P., 1965, Government debt in a neoclassical growth model, Amer-

ican Economic Review 55, 1126-1150.

31

[15] Dollar, D, and R. Gatti, 1999, Gender inequality, income, and growth: Are

good times good for women?, Policy Research Report on Gender and Devel-

opment, Working Paper No. 1, The World Bank.

[16] Echevarria, C, and A. Merlo, 1999, Gender differences in education in a

dynamic bargaining model, International Economic Review 40 (2), 265-286.

[17] Echevarria, C, and K. Moe, 2000, On the need for gender in dynamic models,

Feminist Economics 6 (2), 77-98..

[18] Elul, R., J. Silva-Reus, and O. Volij, 1997, Will you marry me? A perspective

on the gender gap, mimeo, Brown University and University of Alicante.

[19] Filmer, D., E.M. King, and L. Pritchett, 1998, Gender disparity in South

Asia: Comparisons between and within countries, mimeo, The World Bank.

[20] Francois, P., 1998, Gender discrimination without gender difference: theory

and policy responses, Journal of Public Economics 68, 1-32.

[21] Galor, O., and D.N. Weil, 1996, The gender gap, fertility, and growth, Amer-

ican Economic Review 86, 374-387.

[22] Galor, O., and D.N. Weil, 1998, Population, technology, and growth: From

the Malthusian regime to the demographic transition, Brown University

Working Paper No. 98-3.

[23] HadÞeld, G.K., 1999, A coordination model of the sexual division of labor,

Journal of Economic Behavior and Organization 40, 125-153.

32

[24] Heckman, J.H., and J.R. Walker, 1990, The relationship between wages and

income and the timing and spacing of births: evidence from Swedish longi-

tudinal data, Econometrica 58 (6), 1411-1441.

[25] Hill, M.A., and E.M. King, 1995, Women�s education and economic well-

being, Feminist Economics 1, 21-46.

[26] Klasen, S., 2002, Low schooling for girls, slower growth for all? Cross-country

evidence on the effect of gender inequality in education on economic devel-

opment, World Bank Economic Review 16, 345-373.

[27] Knowles, S., P.K. Lorgelly, and P.D. Owen, 2001, Are educational gender gaps

a brake on economic development? Some cross-country empirical evidence,

Oxford Economic Papers, forthcoming.

[28] Lagerlof, N.-P., 2003, Gender equality and long-run growth, Journal of Eco-

nomic Growth, forthcoming.

[29] Nerlove, M., A. Razin, and E. Sadka, 1984, Bequests and the size of popu-

lation when population is endogenous, Journal of Political Economy 92 (31),

527-531.

[30] Olsen, R., 1994, Fertility and the size of the U.S. labor force, Journal of

Economic Literature 32, 60-100.

[31] Osborne, M.J., and A. Rubinstein, 1994, A course in game theory, the MIT

Press, Cambridge, Massachusetts.

33

[32] Sen, A., 1997, Population policy: Authoritarianism versus cooperation, Jour-

nal of Population Economics 10, 3-22.

[33] Schultz, T.P., 1985, Changing world prices, women�s wages, and the fertility

transition: Sweden, 1860-1910, Journal of Political Economy 93 (6), 1126-

1154.

[34] Stokey, N., 1994, Comments on Barro and Lee, Carnegie-Rochester Confer-

ence Series on Public Policy 40, 47-57.

[35] Summers, L.H, 1994, Investing in all the people - educating women in devel-

oping countries, EDI Seminar Paper No. 45, The World Bank.

[36] Summers, R., and A. Heston, 1991, The Penn World Table (mark 5): an

expanded set of international comparisons 1950-1988, Quarterly Journal of

Economics 106, pp.327�368.

[37] United Nations Development Programme, 2000, Human Development Re-

port, United Nations Publications, New York.

34

);,( µσ mtt hn

tn

);( µω tn

++++ µµµµµµµµ

1

1

b

++++

++++++++−−−−

====

µµµµµµµµ

δδδδααααββββαααα

1

1

bn*

tw

Figure 1: Determining wage and fertility rates for non-growing economies.

35

n

);( µω n

pn

++++++++ββββ

δδδδαααα1

)';( µω n

++++

++++++++−−−−

====

µµµµµµµµ

δδδδααααββββαααα

1

1

bn *

++++ µµµµµµµµ

1

1

b

Figure 2: Effects on steady-state fertility when gender equality increases in non-

growing economies.

36

µµµµ big

µµµµ small

1

1++++tγγγγ

*γγγγ

(((( ))))0mh

mth

Figure 3: Stability.

37